We show in this paper how numerical bifurcation analysis can be used to study the evolution of genetically transmitted phenotypic traits. For this, we consider the standard Rosenzweig-MacArthur prey-predator model and, following the so-called Adaptive Dynamics approach, we derive from it a second-order evolutionary model composed of two ODEs, one for the prey trait and one for the predator trai...

We establish rigorous scaling laws for the average bursting time for bubbling bifurcations of an invariant manifold, assuming the dynamics within the manifold to be uniformly hyperbolic. This type of global bifurcation appears in nearly synchronized systems, and is conjectured to be typical among those breaking the invariance of an asymptotically stable hyperbolic invariant manifold. We conside...

In this paper, we consider a simple discrete-time single-directional network of four neurons. The characteristics equation of the linearized system at the zero solution is a polynomial equation involving very high-order terms. We first derive some sufficient and necessary conditions ensuring that all the characteristic roots have modulus less than 1. Hence, the zero solution of the model is asy...

We study the codimension-one and -two bifurcations of the Ornstein–Zernike equation with hypernetted chain (HNC) closure with Lennard–Jones intermolecular interaction potential. The main purpose of the paper is to present the results of a numerical study undertaken using a suite of algorithms implemented in MATLAB and based on pseudo arc-length continuation for the codimension-one case and a Ne...

We reveal and give a theoretical explanation for spiral-like structures of periodicity hubs in the biparameter space of a generic dissipative system. We show that organizing centers for "shrimp"-shaped connection regions in the spiral structure are due to the existence of Shilnikov homoclinics near a codimension-2 bifurcation of saddle-foci.

Vanderbauwhede and van Gils, Krupa, and Langford studied unfoldings of bifurcations with purely imaginary eigenvalues and a nonsemisimple linearization, which generically occurs in codimension three. In networks of identical coupled ODE these nilpotent Hopf bifurcations can occur in codimension one. Elmhirst and Golubitsky showed that these bifurcations can lead to surprising branching patterns...

We describe a codimension-3 bifurcation surface in the space of C-smooth (r ≥ 3) dynamical systems (with phase space of dimension 4 or higher) having an attractive two-dimensional invariant manifold with an infinite sequence of periodic orbits of alternating stability which converge to a homoclinic loop. 2000 Math. Subj. Class. 37G20, 34C07, 34C37.

The purpose of this paper is to present qualitative and bifurcation analysis near the degenerate equilibrium in models of interactions between lymphocyte cells and solid tumor and to understand the development of tumor growth. Theoretical analysis shows that these cancer models can exhibit Bogdanov-Takens bifurcation under sufficiently small perturbation of the system parameters whether it is v...

We analyse the topological (knot-theoretic) features of a certain codimension-one bifurcation of a partially hyperbolic xed point in a ow onR 3 originally described by Shil'nikov. By modifying how the invariant manifolds wrap around themselves, or \pleat," we may apply the theory of templates, or branched two-manifolds, to capture the topology of the ow. This analysis yields a class of ows whic...

We perform a bifurcation analysis of the mathematical model of Jones and Kompala [K.D. Jones, D.S. Kompala, Cybernetic model of the growth dynamics of Saccharomyces cerevisiae in batch and continuous cultures, J. Biotechnol. 71 (1999) 105-131]. Stable oscillations arise via Andronov-Hopf bifurcations and exist for intermediate values of the dilution rate as has been noted from experiments previ...